Terms

Point Estimate is a single value calculated from a sample of data that is used to estimate the value of an unknown parameter of a population.

Confidence Interval is a range of values that likely contains the true population parameter with a certain level of confidence.

Level of Confidence is a probability that the confidence interval actually captures a true value.

Bootstrapping

This is a technique used to estimate the sampling distribution of a statistic by repeatedly re-sampling from the observed data with replacement

Bootstrap Example

This is our sample.

\[(-5,0)*--*-*-*-*---(0,0)--\stackrel{\mu}{|}--*--*--*-*-- (5,0)\]

Then randomly pull 3 values, which becomes our data.

\[(-5,0)---*-----(0,0) \stackrel{\mu}{|}----*-----*-- (5,0)\]

Plot our mean then throw those values back in (i.e., replacement) and pull another 3.

\[(-5,0)--------(0,0)----*--*-\stackrel{\mu}{|}-*---- (5,0)\]

Plot that mean and repeat!

Bootstrap Example

Eventually we will have tons of means (we only have 9 in our example here) but in reality we would have hundreds or even thousands huddled around a similar point on our number line.

\[(-5,0)-\stackrel{\mu}{|}----\stackrel{\mu}{|}---(0,0)---\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}---\stackrel{\mu}{|}- (5,0)\]

Once we have these, we have robust estimates of a true mean within a sample.

Confidence Intervals

Well, when we bootstrapped we essentially created an empirical sampling distribution.

\[(-5,0)-\stackrel{\mu}{|}----\stackrel{\mu}{|}---(0,0)---\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}---\stackrel{\mu}{|}- (5,0)\]

And as a result we can identify the percentiles that corresponds to your desired confidence level.

These percentiles from the bootstrap replicates become the lower and upper bounds of what we will call a confidence interval.

Confidence Intervals

Now that we have this sampling distribution we can calculate our 95% confidence interval.

Which simply means, we have a 95% certainty that our population mean will fall between the two red lines.

And 5% of the time, our population mean could fall outside of that zone.

\[(-5,0)-\stackrel{\mu}{|}--\stackrel{\mu}{|}---(0,0)---\color{red}\vert-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\stackrel{\mu}{|}-\color{red}\vert--\stackrel{\mu}{|}- (5,0)\]

Confidence Interval

\[\text{CI} = \bar{x} \pm z \frac{s}{\sqrt{n}}\]

\(\bar{x}\) = sample mean

\(z\) = confidence level

\(s\) = sample standard deviation

\(n\) = sample size

Confidence Levels

Confidence Level Example

A researcher wants to estimate the average number of hours per week spent by employees in a certain company on remote work activities.

They take a random sample of 80 employees and find that the average number of hours spent on remote work per week in the sample is 12 hours, with a standard deviation of 2 hours.

Calculate the 95% confidence interval for the average number of hours spent on remote work activities per week by all employees in the company.

Confidence Interval Example

Confidence Interval for Small Samples

With smaller samples (n< 30) the Central Limit Theorem does not apply,so we use a different distribution called the t distribution.

In order to locate our t-value, we have to locate the degrees of freedom.

Degrees of Freedom

The number of independent values that are simple allowed to vary within the calculation of a parameter.

\[df = n - 1\]

n = sample size

But why n-1?

n-1 with respect to Degrees of Freedom

Let’s consider an example where 4 people are asked to pick a number that, when combined with the other members, will equal 50.

• Person 1 chooses 10

• Person 2 chooses 20

• Person 3 chooses 10

• Person 4 chooses…… Oof they can’t choose anything! They are no longer free to vary.

• They have to choose 10 because the we want a sum that equals 50.

Confidence Interval for Small Samples

Once we have a df and our confidence level, we head to page 373 in our book or head to this link.

Confidence Interval for Small Samples Example

A biology class wants to estimate the average length of a certain species of fish in a lake.

They take a random sample of 15 fish and measure their lengths. The sample mean length is 30 cm, and the sample standard deviation is 4 cm.

Calculate the 95% confidence interval for the average length of this species of fish in the lake.

Confidence Interval Example

That’s all folks! Have a great weekend!